# Union with Empty Set

## Theorem

The union of any set with the empty set is the set itself:

$S \cup \O = S$

## Proof 1

 $\displaystyle S$ $\subseteq$ $\displaystyle S$ Set is Subset of Itself $\displaystyle \O$ $\subseteq$ $\displaystyle S$ Empty Set is Subset of All Sets $\displaystyle \leadsto \ \$ $\displaystyle S \cup \O$ $\subseteq$ $\displaystyle S$ Union is Smallest Superset $\displaystyle \leadsto \ \$ $\displaystyle S$ $\subseteq$ $\displaystyle S \cup \O$ Set is Subset of Union $\displaystyle \leadsto \ \$ $\displaystyle S \cup \O$ $=$ $\displaystyle S$ Definition of Set Equality

$\blacksquare$

## Proof 2

$\O \subseteq S$
$S \cup \O = S$

$\blacksquare$